Difference between revisions of "2014 AIME II Problems/Problem 14"
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<math>PD=2ND</math> and <math>AP=AD-PD=2HD-2ND=2HN</math> or <math>AP=2HN=HM</math> | <math>PD=2ND</math> and <math>AP=AD-PD=2HD-2ND=2HN</math> or <math>AP=2HN=HM</math> | ||
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+ | Now we know that HM=AP, we can find for HM which is simpler to find. | ||
+ | |||
+ | We can use point B to split it up as HM=HB+BM, | ||
+ | |||
+ | We can chase those lengths and we would get | ||
+ | |||
+ | <math>AB=10</math>, so <math>OB=5</math>, so <math>BC=5\sqrt{2}</math>, so <math>BM=\dfrac{1}{2} \cdot BC=\dfrac{5\sqrt{2}}{2}</math> |
Revision as of 21:39, 29 March 2014
14. In △ABC, AB=10, ∠A=30∘, and ∠C=45∘. Let H, D, and M be points on the line BC such that AH⊥BC, ∠BAD=∠CAD, and . Point is the midpoint of the segment , and point is on ray such that PN⊥BC. Then , where and are relatively prime positive integers. Find .
http://www.artofproblemsolving.com/Wiki/images/5/59/AOPS_wiki.PNG ( This is the diagram.)
As we can see,
is the midpoint of and is the midpoint of
is a triangle, so ∠HAB=15∘.
is .
and are parallel lines so is also.
Then if we use those informations we get and
and or
Now we know that HM=AP, we can find for HM which is simpler to find.
We can use point B to split it up as HM=HB+BM,
We can chase those lengths and we would get
, so , so , so